Problem: Solve for $x$ : $ 4|x - 3| - 8 = -4|x - 3| + 3 $
Explanation: Add $ {4|x - 3|} $ to both sides: $ \begin{eqnarray} 4|x - 3| - 8 &=& -4|x - 3| + 3 \\ \\ { + 4|x - 3|} && { + 4|x - 3|} \\ \\ 8|x - 3| - 8 &=& 3 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 8|x - 3| - 8 &=& 3 \\ \\ { + 8} &=& { + 8} \\ \\ 8|x - 3| &=& 11 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x - 3|} {{8}} = \dfrac{11} {{8}} $ Simplify: $ |x - 3| = \dfrac{11}{8}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 3 = -\dfrac{11}{8} $ or $ x - 3 = \dfrac{11}{8} $ Solve for the solution where $x - 3$ is negative: $ x - 3 = -\dfrac{11}{8} $ Add ${3}$ to both sides: $ \begin{eqnarray} x - 3 &=& -\dfrac{11}{8} \\ \\ {+ 3} && {+ 3} \\ \\ x &=& -\dfrac{11}{8} + 3 \end{eqnarray} $ Change the ${ + 3}$ to an equivalent fraction with a denominator of $8$ $ x = - \dfrac{11}{8} {+ \dfrac{24}{8}} $ $ x = \dfrac{13}{8} $ Then calculate the solution where $x - 3$ is positive: $ x - 3 = \dfrac{11}{8} $ Add ${3}$ to both sides: $ \begin{eqnarray} x - 3 &=& \dfrac{11}{8} \\ \\ {+ 3} && {+ 3} \\ \\ x &=& \dfrac{11}{8} + 3 \end{eqnarray} $ Change the ${ + 3}$ to an equivalent fraction with a denominator of $8$ $ x = \dfrac{11}{8} {+ \dfrac{24}{8}} $ $ x = \dfrac{35}{8} $ Thus, the correct answer is $x = \dfrac{13}{8} $ or $x = \dfrac{35}{8} $.